QUANTUM STOCHASTIC INTEGRALS AS OPERATORS ANDRZEJ L UCZAK Abstract. We constructquantumstochasticintegralsforthe in- 0 tegrator being a martingale in a von Neumann algebra, and the 1 integrand—asuitable processwith valuesinthe samealgebra,as 0 densely defined operators affiliated with the algebra. In the case 2 of a finite algebra we allow the integrator to be an L2–martingale n in which case the integrals are L2–martingales too. a J 2 1 Introduction ] A Thetheoryof‘general’quantumstochasticintegrals(i.e. notfounded F . on Fock space) deals mainly with the setup which can be roughly de- h scribed as follows: For a von Neumann algebra A with a filtration t a {A : t > 0} we have a corresponding process (X(t) : t > 0) with val- m t ues in Lp(A), and a corresponding process (f(t) : t > 0) with values [ in Lq(A), 1/p+1/q = 1/2, 2 6 p, q 6 +∞, Lp(A) and Lq(A) being 1 appropriate noncommutative Lp-spaces. Then under various specific v 9 assumptions about X and f, among which the most natural is that X 5 b b isa martingale, one candefine stochastic integrals f dX and dXf 9 a a 1 as elements of L2(A) (cf. [1, 4, 5, 7, 8, 9, 10]). TRhe advantageRof this . 1 is that, for any reasonable definition of the integral, it may be approx- 0 imated by integral sums of the form f(t )[X(t ) − X(t )] (for 0 i−1 i i−1 b 1 f dX, the other one being definedPin complete analogy). Now a a v: pRroduct of elements from Lp(A) and Lq(A), is an element from Lr(A), i where 1/p+1/q = 1/r. In particular, the approximation holds in the X case r = 2 (in a Hilbert space). However, this approach can be ex- r a ploited also in the following quite natural setting. If we assume that the algebra A acts in a Hilbert space H, and if we let f and X take their values in A, then the integral sum belongs to A too, and we may ask aboutitsbehavior onelements of H. This againleadsus to approx- imation of the integral evaluated at some points of H, i.e., we come to b b the notion of integrals f dX and dXf as operators on H. This a a idea has been carried oRut in [3] forRa particular class of martingales 1991 Mathematics Subject Classification. Primary: 81S25;Secondary: 46L53. Key words and phrases. Quantum stochastic integrals, quantum martingales, adapted processes. Work supported by KBN grant 2 P03A 03024. 1 2 ANDRZEJL UCZAK X defined ‘canonically’ in quasi-free representations of the CCR and CAR. Itturnsoutthatitispossibletoapplythispointofviewinthefollow- ingsituation: Astheintegratorwetakeamonotoneornorm-continuous A-valued process. Then we show that there exists a Riemann-Stieltjes b type integral f dX as a densely defined operator affiliated with the a algebra A. InRthe case when A is finite, we show the existence of both b b the integrals f dX and dXf as densely defined closed operators a a affiliated withR A. MoreovRer, we can weaken the assumptions about (X(t)) and allow it to be a martingale in L2(A), in which case the integrals above will be elements from L2(A) too. Finally, let us say a few words about quantum stochastic integrals in general. In the existing theories of quantum integration, especially those of Lebesgue type, the classes of ‘theoretically admissible’ inte- grands are rather narrow and lack any concrete examples of processes that can be integrated. On the other hand for integrals of Riemann- Stieltjes type such examples have been provided, and, for example, it was shown how one can integrate predictable processes and how inte- gration with respect to a quantum random time can be performed (cf. [8,9]). Inthisnotewetakeasimilarapproachshowingthepossibilityof integrating either a monotone or norm-continuous process with respect to a martingale. It is worth noting that in the case of a monotone pro- cess, apparently nothing can be said about the existence of Lebesgue type integrals, while in the case of a norm-continuous process its exis- tence can be proved only under some additional assumption. 1. Preliminaries and notation Throughout the paper, we assume that A is a von Neumann algebra acting in a Hilbert space H with inner product h·,·i, and that ω is a normal faithful state given by a cyclic and separating unit vector Ω in H, i.e., A is represented in standard form. We suppose, further, that we have an increasing family {A : t ∈ [0,+∞)} of von Neumann t subalgebras of A (A ⊂ A for s 6 t), called a filtration, and a corre- s t sponding family {E } of normal conditional expectations from A onto t A leaving ω invariant. t A process in A or L2(A) is a function defined on [0,+∞) with val- ues in A or L2(A), respectively. We shall denote by f, processes in A, and by X, processes either in A or L2(A). Following the no- tation of probability theory, we shall sometimes denote a process by (X(t) : t > 0) (a family of ‘random variables’), and the same applies to f. The norms in H and L2(A) will be denoted k·k and k·k , respec- H 2 tively, while k·k will stand for the operator norm. QUANTUM STOCHASTIC INTEGRALS 3 Define on (a dense subspace of) H operators P given by: t (1) P (xΩ) = (E x)Ω, x ∈ A. t t It is well–known (cf. e.g. [6] Propositions 1.1 and 1.2) that (P : t > 0) is an increasing family of orthogonal projections in H with t ranges H = A Ω. Moreover, P ∈ A′, where A is the commutant of t t t t t A . t A process in A or L2(A) will be called adapted if its value at each point t > 0 belongs to A or L2(A ), respectively. We call processes f t t in A, and X in L2(A) martingales if for any s,t ∈ [0,+∞), s 6 t, the equalities E f(t) = f(s), E X(t) = X(s) s s hold, where in the L2-case we use the same symbol E to denote the s extension of the conditional expectation to L2(A). It follows that a martingale is an adapted process. Let (X(t) : t ∈ [0,+∞)) be a process, and let 0 6 t 6 t 6 ··· 6 0 1 t < +∞ be a sequence of points. To simplify the notation we put m ∆X(t ) = X(t )−X(t ), k = 1,...,m. k k k−1 Let (X(t) : t ∈ [0,+∞)), (f(t) : t ∈ [0,+∞)) be arbitrary processes, and let [a,b] be a subinterval of [0,+∞). For a partition θ = {a = t < t < ··· < t = b} of [a,b] we form left and right 0 1 m integral sums m m Sl = ∆X(t )f(t ), Sr = f(t )∆X(t ). θ k k−1 θ k−1 k X X k=1 k=1 If there exist limits (in any sense) of the above sums as θ is refined, we call them respectively the left and right stochastic integrals of f with respect to (X(t)), and denote b b limSl = dXf, limSr = f dX. θ θ θ Z θ Z a a This notion of integral is a weaker one than defining the integrals as the limits b b dXf = lim Sl, f dX = lim Sr, θ θ Za kθk→0 Za kθk→0 where kθk stands for the mesh of the partition θ. A definition of this kind is standard in the classical theory of Riemann-Stieltjes as well as the theory of stochastic integrals. It is worth noticing that in non- commutative integration theory, whenever this Riemann-Stieltjes type integral is considered, its definition refers to the weaker form of the limit with the refining net of partitions (cf. [2, 8, 9]). However, un- der additional assumptions we shall be able to obtain the integral also in the stronger sense thus making it similar to the classical stochastic integral. 4 ANDRZEJL UCZAK 2. Integrals as operators on H — the non-tracial case Our construction of the integral is given by the following Theorem 1. Let (X(t) : t > 0) be a martingale in A, and let f: [0,∞) → A be a hermitian adapted process such that f is mono- tone. Then for each t > 0 there exists a Riemann–Stieltjes type integral tf dX, which is a densely defined operator on H affiliated with A. 0 R Proof. Fix t > 0,and let θ = {0 = t < ... < t = t} be a partition of 0 m [0,t]. Put m Sr = f(t )[X(t )−X(t )]. θ k−1 k k−1 X k=1 We want to define tf dX as an operator on A′Ω (where A′ is the 0 commutant of A) byR t f dX(x′Ω) = limSr(x′Ω), x′ ∈ A′, Z θ θ 0 as θ is refined. For this it is sufficient to show the existence of the limit lim SrΩ, since θ θ Srx′ = x′Sr. θ θ We have m SrΩ = f(t )[X(t )−X(t )]Ω θ k−1 k k−1 X k=1 m = f(t )[E X(t)−E X(t)]Ω k−1 tk tk−1 X k=1 m = f(t )(P −P )X(t)Ω, k−1 tk tk−1 X k=1 where the P are projections defined by (1). Consider the operator sum t m m (2) σr = f(t )(P −P ) = (P −P )f(t ) = (σr)∗. θ k−1 tk tk−1 tk tk−1 k−1 θ X X k=1 k=1 Tofixattention, assumethatf isincreasing(i.e., f(s) 6 f(t)fors 6 t), and let θ′ = θ ∪{t′} for some t < t′ < t be a one-point refinement j j+1 of θ. Then σθr′ −σθr = f(tj)(Pt′ −Ptj)+f(t′)(Ptj+1 −Pt′)−f(tj)(Ptj+1 −Ptj) = [f(t′)−f(tj)](Ptj+1 −Pt′) = (Ptj+1 −Pt′)[f(t′)−f(tj)](Ptj+1 −Pt′) > 0, because Ptj+1−Pt′ is a projection commuting with f(tj) and f(t′), and f(t′) −f(t ) > 0. It follows that the net {σr} is increasing. Further- j θ more, for each k = 1,...,m we have (P −P )f(t )2(P −P ) 6 kf(t )k2(P −P ). tk tk−1 k−1 tk tk−1 k−1 tk tk−1 QUANTUM STOCHASTIC INTEGRALS 5 Hence m m (P −P )f(t )2(P −P ) 6 kf(t )k2(P −P ) tk tk−1 k−1 tk tk−1 k−1 tk tk−1 X X k=1 k=1 m 6 c2 (P −P ) = c2(P −P ), tk tk−1 t 0 X k=1 where c = sup kf(s)k = max{kf(0)k,kf(t)k}. Consequently, 06s6t m kσrk2 = k(σr)2k = (P −P )f(t )f(t )(P −P ) θ θ tj tj−1 j−1 i−1 ti ti−1 (cid:13)(cid:13)iX,j=1 (cid:13)(cid:13) m (cid:13) (cid:13) = (P −P )f(t )2(P −P ) 6 c2kP −P k = c2, ti ti−1 i−1 ti ti−1 t 0 (cid:13)(cid:13)Xi=1 (cid:13)(cid:13) (cid:13) (cid:13) which means that {σr} is norm-bounded. This, together with the fact θ that the net is increasing, yields the existence of lim σr in the strong θ θ operator topology, in particular, there exists limσr(X(t)Ω) = limSrΩ. θ θ θ θ It is clear that for each x′,y′ ∈ A′ we have t f dX y′(x′Ω) = limSry′(x′Ω) = (cid:18)Z (cid:19) θ θ 0 t y′limSr(x′Ω) = y′ f dX (x′Ω), θ θ (cid:18)Z (cid:19) 0 thus tf dX is affiliated with A′′ = A. (cid:3) 0 R It turns out that even a stronger form of integral can be obtained for norm-continuous processes. Theorem 2. Let (X(t) : t > 0) be a martingale in A, and let f: [0,∞) → A be a norm-continuous adapted process. Then for each t > 0 there exists a Riemann-Stieltjes type integral tf dX which is a 0 densely defined operator on H affiliated with A. MorReover, this integral is given as the limit m lim f(t )∆X(t ) k−1 k kθk→0X k=1 on A′Ω. Proof. Fix t > 0. We shall show that the net {SrΩ} is Cauchy as the θ mesh kθk of the partition θ tends to 0. Take an arbitrary ε > 0, and let δ > 0 be such that for each t′,t′′ ∈ [0,t] with |t′ −t′′| < δ we have ε kf(t′)−f(t′′)k < . 2kX(t)Ωk H 6 ANDRZEJL UCZAK Let θ′ = {0 = t < t < ··· < t = t} be an arbitrary partition of [0,t] 0 1 m with kθ′k < δ, and let θ′′ be a partition of [0,t] finer than θ′. Denote by t(k),t(k),...,t(k) the points of θ′′ lying between t and t , such that 0 1 lk k−1 k (k) (k) (k) t = t < t < ··· < t = t . We then have k−1 0 1 lk k m lk Sr = f(t(k))∆X(t(k)) θ′′ i−1 i Xk=1Xi=1 m m lk Sr = f(t )∆X(t ) = f(t )∆X(t(k)), θ′ k−1 k k−1 i Xk=1 Xk=1Xi=1 so that m lk Sr −Sr = [f(t(k) )−f(t )]∆X(t(k)). θ′′ θ′ i−1 k−1 i Xk=1Xi=1 As in the proof of Theorem 1 we have m lk (Sr −Sr)Ω = f(t(k))−f(t ) ∆X(t(k))Ω θ′′ θ′ i−1 k−1 i Xk=1Xi=1 h i m lk (k) = f(t )−f(t ) (P −P )X(t)Ω Xk=1Xi=1 h i−1 k−1 i t(ik) t(i−k)1 =(σr −σr )X(t)Ω, θ′′ θ′ and thus m lk k(Sr −Sr)Ωk2 = k(P −P ) f(t(k))−f(t ) X(t)Ωk2 θ′′ θ′ 2 Xk=1Xi=1 t(ik) t(i−k)1 (cid:2) i−1 k−1 (cid:3) H by the orthogonality of P −P and P −P for i 6= j. Further- more t(ik) t(i−k)1 t(jk) t(jk−)1 k(P −P ) f(t(k) )−f(t ) X(t)Ωk2 t(ik) t(i−k)1 i−1 k−1 H (cid:2) (cid:3) =h(P −P )|f(t(k))−f(t )|2(P −P )X(t)Ω,X(t)Ωi, t(ik) t(i−k)1 i−1 k−1 t(ik) t(i−k)1 (k) and since |t −t | < δ, we have i−1 k−1 ε2 |f(t(k))−f(t )|2 6 1, i−1 k−1 4kX(t)Ωk2 H giving ε2 (P −P )|f(t(k))−f(t )|2(P −P ) 6 (P −P ). t(ik) t(i−k)1 i−1 k−1 t(ik) t(i−k)1 4kX(t)Ωk2H t(ik) t(i−k)1 QUANTUM STOCHASTIC INTEGRALS 7 This yields the estimate k(P −P ) f(t(k))−f(t ) X(t)Ωk2 t(ik) t(i−k)1 i−1 k−1 H (cid:2) (cid:3) =h(P −P )|f(t(k))−f(t )|2(P −P )X(t)Ω,X(t)Ωi t(ik) t(i−k)1 i−1 k−1 t(ik) t(i−k)1 ε2 6 k(P −P )X(t)Ωk2 . 4kX(t)Ωk2H t(ik) t(i−k)1 H Consequently, we obtain (3) m lk k(Sr −Sr)Ωk2 = k(P −P ) f(t(k) )−f(t ) X(t)Ωk2 θ′′ θ′ H Xk=1Xi=1 t(ik) t(i−k)1 (cid:2) i−1 k−1 (cid:3) H ε2 m lk 6 k(P −P )X(t)Ωk2 4kX(t)Ωk2H Xk=1Xi=1 t(ik) t(i−k)1 H ε2 m lk 2 = (P −P )X(t)Ω 4kX(t)Ωk2H(cid:13)(cid:13)Xk=1Xi=1 t(ik) t(i−k)1 (cid:13)(cid:13)H ε2 (cid:13) (cid:13) = k(P −P )X(t)Ωk2 4kX(t)Ωk2 t 0 H H ε2 ε2 6 kX(t)Ωk2 = . 4kX(t)Ωk2 H 4 H Let now θ and θ be arbitrary partitions of [0,t] such that 1 2 kθ k < δ, kθ k < δ, and let θ′′ = θ ∪θ . Then we have by (3) 1 2 1 2 ε ε k(Sr −Sr )Ωk < and k(Sr −Sr )Ωk < , θ′′ θ1 H 2 θ′′ θ2 H 2 so k(Sr −Sr )Ωk < ε, θ1 θ2 H showing that the net {SrΩ} is Cauchy. Thus there exists lim SrΩ θ kθk→0 θ and the rest of the proof is the same as that of Theorem 1. (cid:3) 3. Integrals as operators on H — the tracial case Let us now assume that ω is a normal tracial state. Recall that the Lebesgue space L2(A,ω) is formally defined as the completion of A with respect to the norm kxk = [ω(x∗x)]1/2 = kxΩk , 2 H and may be realized as a space of densely defined closed operators affiliated with A such that Ω belongs to their domains. For the von Neumann subalgebras A the normal ω-invariant con- t ditional expectations E : A → A extend to orthogonal projections t t 8 ANDRZEJL UCZAK (denoted by the same letter) from L2(A,ω) onto L2(A ,ω). If we de- t fine an operator P on H by t (4) P (XΩ) = (E X)Ω, X ∈ L2(A,ω), t t then P is an orthogonal projection from A′. (The definition above is t t thesameasthatgivenby (1)forX ∈ A.) Foranyx ∈ A, A ∈ L2(A,ω) the operators xA and Ax belong to L2(A,ω), consequently we may again consider the integral sums m m f(t )∆X(t ), ∆X(t )f(t ), k−1 k k k−1 X X k=1 k=1 where f: [0,∞) → A, X: [0,∞) → L2(A,ω), as operators on H. We shall use the notation m m Sr(f,X) = f(t )∆X(t ), Sl(f,X) = ∆X(t )f(t ), θ k−1 k θ k k−1 X X k=1 k=1 for the integral sums, to indicate their dependence on f and X. Then since Sr(f,X) and Sl(f,X) are affiliated with A, we have an explicit θ θ description of their actions on A′Ω as Sr(f,X)(x′Ω) = x′Sr(f,X)Ω, Sl(f,X)(x′Ω) = x′Sl(f,X)Ω. θ θ θ θ Theorem 3. Let (X(t) : t > 0) be a martingale in L2(A,ω) and let f: [0,∞) → A be either monotone or norm continuous. Then for each t > 0 there exist integrals tf dX and tdXf as elements of 0 0 L2(A,ω). Moreover, the L2(A,ω)R-processes (YR(t) : t > 0), (Z(t) : t > 0) defined by t t Y(t) = dXf, Z(t) = f dX Z Z 0 0 are martingales. Proof. The existence of the integral tf dX is proved exactly as in 0 Theorems 1 and 2 upon observing thRat according to formula (4) we have [X(t )−X(t )]Ω = E X(t)−E X(t) Ω = (P −P )X(t)Ω. k k−1 tk tk−1 tk tk−1 It follows that the net {(cid:2)Sr(f,X)Ω} is Cauch(cid:3)y, and since θ kSr(f,X)k = kSr(f,X)k , θ H θ 2 {Sr(f,X)} converges in k·k -norm, and thus its limit is an element of θ 2 L2(A,ω). For Sl(f,X) we have θ Sl(f,X) = [Sr(f∗,X∗)]∗; θ θ so we obtain kSl(f,X)k = k[Sr(f∗,X∗)]∗k = kSr(f∗,X∗)k , θ 2 θ 2 θ 2 QUANTUM STOCHASTIC INTEGRALS 9 because Ω is tracial. Since f∗ satisfies the same assumptions as f, and (X(t)∗ : t > 0) is also an L2(A,ω)–martingale, we obtain the convergence of {Sl(f,X)} in k·k -norm. θ 2 Now we shall show that (Y(t) : t > 0) is a martingale. Fix t > 0 and take an arbitrary s < t. We have t dXf = lim Sl. θ Z0 kθk→0 We may assume that s is one of the points of each partition θ = {0 = t < t < ··· < t = t}, say s = t . Then we have 0 1 m k k m E Sl = E ( [X(t )−X(t )]f(t )+ [X(t )−X(t )]f(t )) s θ s i i−1 i−1 i i−1 i−1 Xi=1 i=Xk+1 k m = E [X(t )−X(t )]f(t )+ E [X(t )−X(t )]f(t ). s i i−1 i−1 s i i−1 i−1 Xi=1 i=Xk+1 For i 6 k we have t 6 s, and thus i E [X(t )−X(t )]f(t ) = [X(t )−X(t )]f(t ), s i i−1 i−1 i i−1 i−1 while for i > k we have t > s, and thus i−1 E [X(t )−X(t )]f(t ) = E E [X(t )−X(t )]f(t ) s i i−1 i−1 s ti−1 i i−1 i−1 =E (E [X(t )−X(t )])f(t ) = 0 s ti−1 i i−1 i−1 by the martingale property. Consequently, k (5) E Sl = [X(t )−X(t )]f(t ). s θ i i−1 i−1 Xi=1 But the sum on the right hand side of (5) is an integral sum for the s integral dXf, and passing to the limit in (5) yields 0 R t s E dXf = dX f, s Z Z 0 0 which shows that (Y(t)) is a martingale. Analogously for (Z(t) : t > 0). (cid:3) Remarks. 1. The formulas t t f dX (x′Ω) = x′ limSrΩ, dXf (x′Ω) = x′ limSlΩ (cid:18)Z (cid:19) θ θ (cid:18)Z (cid:19) θ θ 0 0 describe explicitly the actions of the integrals on A′Ω. 2. Let us notice that an attempt to define Lebesgue type integrals above, e.g., along the lines of [4] or [10], would be successful only in the simple case of norm-continuous f, and even then under the additional assumption of left- or right-continuity in k·k -norm of the martingale 2 (X(t)). The reason for this is that this type of integral is defined for 10 ANDRZEJL UCZAK µ–measurable functions f ∈ L2([0,∞),µ,A), where µ is a measure defined by µ([a,b)) or µ((a,b]) = ω(|X(b)−X(a)|2) = ω(|X(b)|2)−ω(|X(a)|2). The continuity of the martingale allows then the extension of µ from the intervals to the Borel sets. For the case of monotone f the failure of the definition of the integral as Lebesgue type is most strikingly seen when the function f is increasing projection-valued. To be more concrete, assume that e is a spectral measure with support [0,∞), and put f(t) = e([0,t]). Then for s < t, f(t)−f(s) is a non-zero projection, so kf(t)−f(s)k = 1, which means that f is not norm continuous either from the left or from the right at any point, while a µ–measurable function must be norm-continuous on some compact set. References [1] C.Barnett,S.Goldstein,I.F.Wilde,Quantumstochasticintegrationandquan- tum stochastic differential equations, Math. Proc.CambridgePhilos.Soc. 116 (1994), 535–553. [2] C.Barnett,T.Lyons,Stopping non-commutative processes,Math.Proc.Cam- bridge Philos. Soc. 99 (1986), 151–161. [3] C. Barnett, R.F. Streater, I.F. Wilde, Quasi-free quantum stochastic integrals for the CCR and CAR, J. Funct. Anal. 52 (1983), 19–47. [4] , Stochastic integrals in an arbitrary probability gauge space, Math. Proc. Cambridge Philos. Soc. 94 (1983), 541–551. [5] , Quantum stochastic integrals under standing hypotheses, J. Math. Anal. Appl. 127 (1987), 181–192. [6] C. Barnett, B. Thakrar, A non-commutative random stopping theorem, J. Funct. Anal. 88 (1990), 342–350. [7] C. Barnett, I.F. Wilde, Belated integrals, J. Funct. Anal. 66 (1986), 283–307. [8] , Random times and time projections, Proc. Amer. Math. Soc. 110 (1990), 425–440. [9] , Random times, predictable processes and stochastic integration in fi- nite von Neumann algebras, Proc. London Math. Soc. 67 (1993), 355–383. [10] S. Goldstein, Conditional expectation and stochastic integrals in non- commutative Lp spaces, Math.Proc.CambridgePhilos.Soc. 110(1991),365– 383. Faculty of Mathematics and Computer Science, L o´d´z University, ul. Banacha 22, 90–238 L o´d´z, Poland E-mail address: [email protected]